In 1903, in the fictional town of Fulton Corners, Pike County, Indiana, a young Jeremiah Kincaid, Bobby Driscoll, lives with Granny Kincaid, Beulah Bondi, on her homestead. Jeremiah idolizes the race horse Dan Patch and dreams of owning and raising a champion of his own. These dreams shift when Granny's sheep give birth to a lamb with black wool whose mother rejects him. Taking pity, Jeremiah convinces Granny to keep the lamb as a pet, naming him Danny.
A Wise Old Owl in Jeremiah's scrapbook cards, Ken Carson, reminds him that:
> it's what you do with what you've got that counts
and this guiding line becomes a touchstone for his ambition and perseverance.
Danny soon becomes a nuisance—underfoot and destructive. To Granny's consternation, Jeremiah neglects his chores and becomes single-mindedly preoccupied with Danny. With the encouragement of his Uncle Hiram, [Burl Ives], and his friend Tildy, [Luana Patten], Jeremiah grows more determined to enter Danny at the Pike County Fair and win a blue ribbon. After a particularly destructive rampage, Granny resolves to get rid of Danny, but cannot bring herself to do it seeing how Jeremiah cares for him.
Uncle Hiram attempts to broach the subject of the County Fair to Granny, but Granny refuses on financial grounds. The Wise Old Owl instructs Jeremiah to persevere, with the stories of Christopher Columbus and Robert the Bruce illustrating "stick-to-it-ivity". Eager to make money to change Granny's mind, Jeremiah and Tildy go looking for scarce wild honey to sell to the general store. Their search takes them into a swamp, where they find a tree swarming with bees, which yields enough honey to fill two tubs' worth, earning Jeremiah a considerable amount of money. Upon returning home, Granny announces that Tildy and Danny have gone missing. Jeremiah runs after Danny, and Tildy returns shortly after. When Jeremiah does not return home before dark, Granny goes out into a storm to look for him, bringing him home without Danny over his protests. When Granny advises Jeremiah that Danny is in God's hands, Jeremiah protests that God cannot take Danny away from him. Granny scolds Jeremiah for becoming selfish and fixated on the prizes Danny can win over genuine love for him.
Jeremiah sets out early the next morning and finds Danny. He reveals to Granny that he prayed and promised God that if He let him find Danny, he would give up his vain dreams. Seeing that Jeremiah has learned his lesson, Granny tells a white lie that she, too, had promised God that if Jeremiah found Danny, they could go to the fair, reasoning that since she has known God longer, her promise takes precedence.
At the fair, Granny advises Jeremiah to accept loss as gracefully as victory. Jeremiah falters slightly when faced with the high pedigrees of the competitors, and Danny causes a minor stumble that momentarily unsettles him, but the judge is impressed nonetheless. The judge compliments Jeremiah and Danny, telling the crowd that Danny is in a class of his own, echoing the Owl's lesson, and awards Jeremiah and Danny a special honor. The family returns to Fulton Corners to the admiration of the community, with Danny now accepted and celebrated.
